MathDB

SMT 2022 Algebra #2

Source:

3/27/2023

Find the sum of the solution(s)

x

to the equation

x=\sqrt{2022+\sqrt{2022+x}}.

SMT 2022 Algebra Tiebreaker #2

Source:

3/28/2023

What is the area of the region in the complex plane consisting of all points

z

satisfying both

|\tfrac{1}{z}-1|<1

and

|z-1|<1

? (

|z|

denotes the magnitude of a complex number, i.e.

|a+bi|=\sqrt{a^2+b^2}

.)

SMT 2022 Calculus #2

Source:

3/29/2023

The straight line

y=ax+16

intersects the graph of

y=x^3

at

2

distinct points. What is the value of

a

?

SMT 2022 Calculus Tiebreaker #2

Source:

3/29/2023

Water is flowing out through the smaller base of a hollow conical frustum formed by taking a downwards pointing cone of radius

12\text{m}

and slicing off the tip of the cone in a cut parallel to the base so that the radius of the cross-section of the slice is

6\text{m}

(meaning the smaller base has a radius of

6\text{m}

). The height of the frustum is

10\text{m}

. If the height of the water level in the frustum is decreasing at

3\text{m/s}

and the current height is

5\text{m}

, then the volume of the water in the frustum is decreasing at

d\text{ m}^3\text{/s}

. Compute

d

.

SMT 2022 Discrete #2

Source:

4/1/2023

Call a three-digit number

\overline{ABC}

<span class='latex-italic'>spicy</span>

if it satisfies

\overline{ABC}=A^3+B^3+C^3

. Compute the unique

n

for which both

n

and

n+1

are

<span class='latex-italic'>spicy</span>

.

SMT 2022 Discrete Tiebreaker #2

Source:

4/1/2023

Let

a

,

b

,

c

be the solutions to

x^3+3x^2-1=0

. Define

S_n=a^n+b^n+c^n

. Given that there are integers

0\le i,j,k\le36

such that

S_n\equiv i^n+j^n+k^n\pmod{37}

for all positive integer

n

, determine the product

ijk

.

SMT 2022 Geometry #2

Source:

4/1/2023

Let

ABC

be an acute, scalene triangle. Let

H

be the orthocenter. Let the circle going through

B

,

H

, and

C

intersect

CA

again at

D

. Given that

\angle ABH=20^\circ

, find, in degrees,

\angle BDC

.

SMT 2022 Geometry Tiebreaker #2

Source:

4/1/2023

The incircle of

\triangle ABC

is centered at

I

and is tangent to

BC

,

CA

, and

AB

at

D

,

E

, and

F

, respectively. A circle with radius

2

is centered at each of

D

,

E

, and

F

. Circle

D

intersects circle

I

at points

D_1

and

D_2

. The points

E_1

,

E_2

,

F_1

, and

F_2

are defined similarly. If the inradius of

\triangle ABC

is

5

, what is the ratio of the area of the triangle whose sides are formed by extending

D_1D_2

,

E_1E_2

, and

F_1F_2

to the area of

\triangle ABC

?

2

Problems(8)